2021/08/01 by Jean-Philippe Chassé, Chassé, Jean-Philippe
Mathematics · #32Q65 #53B20 (Secondary) #53C17 #53D12 (Primary) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:32Q65 #msc:53B20 #msc:53C17 #msc:53D12
paper · pdf · doi:10.48550/arxiv.2108.00555
Submitted for consideration in the International Journal of Mathematics; 37 pages, 5 figures. Corrected typos with regards to the references and added a link to read Perelman's 1991 preprint
arxiv created 2021/10/16 · arxiv updated 2021/10/19
We consider collections of Lagrangian submanifolds of a given symplectic manifold which respect uniform bounds of curvature type coming from an auxiliary Riemannian metric. We prove that, for a large class of metrics on these collections, convergence to an embedded Lagrangian submanifold implies convergence to it in the Hausdorff metric. This class of metrics includes well-known metrics such as the Lagrangian Hofer metric, the spectral norm and the shadow metrics introduced by Biran, Cornea and Shelukhin arXiv:1806.06630. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball.